Rogers-Szego Polynomials
Rogers-Szego Polynomials
羅傑斯-斯澤格多項式(英語:Rogers–Szegő polynomials)是1926年匈牙利數學家斯澤格首先研究的在單位圓上的正交多項式,以Q階乘冪定義如下;
![{\displaystyle h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c417ae309ffa3f054dba2cd96d09b99ee53311)
前面幾個羅傑斯-斯澤格多項式為:
![{\displaystyle h_{1}(x;q)=1+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89cc93f9097303d226478485c572cebd3a43108a)
![{\displaystyle h_{2}(x;q)=1+{\frac {(1-q^{2})*x}{(1-q)}}+x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f5ef0d22983a9df96d662efeb4e5caa415a2783)
![{\displaystyle h_{3}(x;q)=1+{\frac {(1-q^{3})*x}{(1-q)}}+{\frac {(1-q^{3})*x^{2}}{(1-q)}}+x^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37f1b2a4a82ec8cea7c0ac93b77853fa287fae69)
![{\displaystyle h_{4}(x;q)=1+{\frac {(1-q^{4})*x}{(1-q)}}+{\frac {(1-q^{3})*(1-q^{4})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*x^{3}}{(1-q)}}+x^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2e30a2c77a9e0d4dd43d70f35eaf337bbace8a)
![{\displaystyle h_{5}(x;q)=1+{\frac {(1-q^{5})*x}{(1-q)}}+{\frac {(1-q^{4})*(1-q^{5})*x^{2}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{4})*(1-q^{5})*x^{3}}{((1-q)*(1-q^{2}))}}+{\frac {(1-q^{5})*x^{4}}{(1-q)}}+x^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee58164944a7664b71c7182d9fe1018f26e3ead0)
- Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574
- Szegő, Gábor, Beitrag zur theorie der thetafunktionen, Sitz Preuss. Akad. Wiss. Phys. Math. Ki., 1926, XIX: 242–252, Reprinted in his collected papers